Question

Use the substitution method to solve the system 2x+3y=4 , −4x−6y=−8 .(1 point)%0D%0AResponses%0D%0A%0D%0A(2−1)%0D%0Aleft parenthesis 2 comma negative 1 right parenthesis%0D%0A%0D%0Ano solution%0D%0Ano solution%0D%0A%0D%0Ainfinite solutions%0D%0Ainfinite solutions%0D%0A%0D%0A(−1,2)

Answer 1

To solve the system of equations using the substitution method, we start with the given equations:

1. \( 2x + 3y = 4 \) (Equation 1)
2. \( -4x - 6y = -8 \) (Equation 2)

Let's isolate \( y \) from Equation 1:

\[
3y = 4 - 2x
\]

\[
y = \frac{4 - 2x}{3}
\]

Now we substitute \( y \) in Equation 2. First, notice that we can simplify Equation 2:

\[
-4x - 6y = -8
\]

Dividing the entire equation by -2 gives us:

\[
2x + 3y = 4
\]

This is exactly the same as Equation 1.

Since both equations represent the same line, there are infinitely many solutions. Any point on the line defined by \( 2x + 3y = 4 \) will satisfy both equations.

Thus, the answer is **infinite solutions**.